Effective Wadge Hierarchy in Computable Quasi-Polish Spaces

We define and study an effective version of the Wadge hierarchy in computable quasi-Polish spaces which include most spaces of interest for computable analysis. Along with hierarchies of sets we study hierarchies of $k$-partitions which are interesting on their own. We show that levels of such hierarchies are preserved by the computable effectively open surjections, that if the effective Hausdorff-Kuratowski theorem holds in the Baire space then it holds in every computable quasi-Polish space, and we extend the effective Hausdorff theorem to $k$-partitions. We establish sufficient conditions for the non-collapse of the effective Wadge hierarchy and apply them to some concrete spaces like the discrete spaces of intergers, Baire space and Cantor. We show that the proof of non-collapse in any concrete space is highly non-trivial.